GCSE N1 - integers, decimals and real numbers

This unit combines skills originally covered by Edexcel's SoW modules: Edexcel module H1.8, Edexcel module H1.9 and Edexcel module H2.1.
The References, Prior Learning, Key Objectives, Differentiation and Notes are drawn together summatively from these Edexcel modules.
Time allocation: 6 to 14 hours
GCSE Scheme of Work Higher skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

			Objectives and approaches	Teaching Resources & Links
10a	A*		pupils determine bounds for calculated quantities from products or sums such as area dividing and subtracting to find LUB and GLB and concluding a reasonable accuracy to report pupils simplify surds by rationalising the denominator where the denominator is of form a ± b√c by using the difference of two squares pupils determine bounds for calculated quantities such as area multiplying and adding to find LUB and GLB and concluding a reasonable accuracy to report pupils simplify surds by rationalising the denominator where the denominator is of form a√b by multiplying by √b pupils are encouraged to challenge their understanding of decimal notation to consider whether the values 3.5 and 3.49• are ‘close’ or ‘equal’ and use this to understand why “A value rounds to 3 to the nearest whole number. What is the maximum it could have been?” is an unanswerable question and hence see the need for defining the LUB instead. pupils use their understanding of bounds in the context of measurement to give least upper bound (LUB, supremum, incorrectly called maximum on some past papers) and greatest lower bound (GLB, infimum, incorrectly called the minimum on p.p.) pupils simplify sums of surds such as √75 - √27 by writing in the form a√c ± b√c and simplifying the result pupils simplify surds by removing square factors such as √75 = 5√3
10b
10c
9a	A		In simplifying algebraic expressions, pupils use rules of indices for negative and fractional values pupils determine the bounds of intervals pupils understand and use rational and irrational numbers
9b
9c
8a	B		pupils solve problems involving calculating with powers and roots pupils solve problems involving calculating with .. numbers expressed in standard form, checking for correct order of magnitude pupils solve problems in standard form (inc. on a calculator pupils sketch and use reciprocal functions
8b
8c
6c

GCSE Scheme of Work Upper Foundation skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

		Objectives and approaches	Teaching Resources & Links
7a	C	pupils solve numerical problems involving multiplication & division with numbers of any size x and with negative numbers in making estimates, pupils round to one significant figure and multiply and divide mentally pupils understand the effects of multiplying and dividing by numbers between 0 & 1
7b
7c
6a	D	pupils…approximate decimals when solving.. equations [ such as x3 + x = 20] using trial and improvement methods pupils use index notation for small positive powers pupils estimate square roots and cube roots using ICT where appropriate
6b
6c
5a	E	pupils… add & subtract negative numbers in context pupils use all four operations with decimals to two decimal places pupils understand and use an appropriate non-calculator method that involve x and by any 3-digit and 2-digit number pupils find, understand and use common factors pupils sort into order directed (positive and negative) numbers pupils use their understanding of place value to multiply and divide whole numbers and decimals by 10, 100 & 1000 pupils check their solutions by… estimating using approximations pupils use all four operations with decimals pupils use brackets appropriately pupils check their solutions by applying inverse operations pupils calculate fractional parts of quantities and measurements using a calculator where appropriate	.
5b
5c

GCSE Scheme of Work Lower Foundation skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

4a	F	in solving number problems, pupils use a range of mental methods of computation with the four operations pupils add and subtract decimals to two decimal places pupils use a range of mental methods including mental recall of multiplication facts up to 10 x 10 and quick derivation of division facts pupils recognise and describe relationships including multiple, factor and square pupils recognise approximate proportions of a whole and use simple fractions and percentages to describe these pupils use their understanding of place value to multiply and divide whole numbers by 10 or 100 in solving problems with or without a calculator pupils check the reasonableness of their results by reference to their knowledge of the context or to the size of the numbers pupils use efficient written methods of addition and subtraction pupils use and understand short multiplication & division methods with a single digit and a 2-digit number (e.g. 38 × 9 and 96 ÷ 4)
4b
4c
3a	G	pupils show understanding of place value in numbers up to 1000 and use this to make approximations. pupils begin to use decimal notation and to recognise negative numbers, in contexts such as money & temperature. pupils use mental recall of addition and subtraction facts to 20 in solving problems involving larger numbers. pupils add and subtract numbers with two digits mentally. pupils add and subtract numbers with three digits using written methods. pupils use mental recall of 2,3,4,5 & 10 multiplication tables and derive division facts pupils solve whole number problems involving x &
3b
3c

GCSE Scheme of Work Higher skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

			Objectives and approaches	Teaching Resources & Links
10a	A*		pupils determine bounds for calculated quantities from products or sums such as area dividing and subtracting to find LUB and GLB and concluding a reasonable accuracy to report pupils simplify surds by rationalising the denominator where the denominator is of form a ± b√c by using the difference of two squares pupils determine bounds for calculated quantities such as area multiplying and adding to find LUB and GLB and concluding a reasonable accuracy to report pupils simplify surds by rationalising the denominator where the denominator is of form a√b by multiplying by √b pupils are encouraged to challenge their understanding of decimal notation to consider whether the values 3.5 and 3.49• are ‘close’ or ‘equal’ and use this to understand why “A value rounds to 3 to the nearest whole number. What is the maximum it could have been?” is an unanswerable question and hence see the need for defining the LUB instead. pupils use their understanding of bounds in the context of measurement to give least upper bound (LUB, supremum, incorrectly called maximum on some past papers) and greatest lower bound (GLB, infimum, incorrectly called the minimum on p.p.) pupils simplify sums of surds such as √75 - √27 by writing in the form a√c ± b√c and simplifying the result pupils simplify surds by removing square factors such as √75 = 5√3
10b
10c
9a	A		In simplifying algebraic expressions, pupils use rules of indices for negative and fractional values pupils determine the bounds of intervals pupils understand and use rational and irrational numbers
9b
9c
8a	B		pupils solve problems involving calculating with powers and roots pupils solve problems involving calculating with .. numbers expressed in standard form, checking for correct order of magnitude pupils solve problems in standard form (inc. on a calculator pupils sketch and use reciprocal functions
8b
8c
6c

GCSE Scheme of Work Upper Foundation skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

		Objectives and approaches	Teaching Resources & Links
7a	C	pupils solve numerical problems involving multiplication & division with numbers of any size x and with negative numbers in making estimates, pupils round to one significant figure and multiply and divide mentally pupils understand the effects of multiplying and dividing by numbers between 0 & 1
7b
7c
6a	D	pupils…approximate decimals when solving.. equations [ such as x3 + x = 20] using trial and improvement methods pupils use index notation for small positive powers pupils estimate square roots and cube roots using ICT where appropriate
6b
6c
5a	E	pupils… add & subtract negative numbers in context pupils use all four operations with decimals to two decimal places pupils understand and use an appropriate non-calculator method that involve x and by any 3-digit and 2-digit number pupils find, understand and use common factors pupils sort into order directed (positive and negative) numbers pupils use their understanding of place value to multiply and divide whole numbers and decimals by 10, 100 & 1000 pupils check their solutions by… estimating using approximations pupils use all four operations with decimals pupils use brackets appropriately pupils check their solutions by applying inverse operations pupils calculate fractional parts of quantities and measurements using a calculator where appropriate	.
5b
5c

GCSE Scheme of Work Lower Foundation skillsfor teaching 2009-2011

Topic reference:

N1

Topic title:

Properties of numbers, calculations in ℤ

4a	F	in solving number problems, pupils use a range of mental methods of computation with the four operations pupils add and subtract decimals to two decimal places pupils use a range of mental methods including mental recall of multiplication facts up to 10 x 10 and quick derivation of division facts pupils recognise and describe relationships including multiple, factor and square pupils recognise approximate proportions of a whole and use simple fractions and percentages to describe these pupils use their understanding of place value to multiply and divide whole numbers by 10 or 100 in solving problems with or without a calculator pupils check the reasonableness of their results by reference to their knowledge of the context or to the size of the numbers pupils use efficient written methods of addition and subtraction pupils use and understand short multiplication & division methods with a single digit and a 2-digit number (e.g. 38 × 9 and 96 ÷ 4)
4b
4c
3a	G	pupils show understanding of place value in numbers up to 1000 and use this to make approximations. pupils begin to use decimal notation and to recognise negative numbers, in contexts such as money & temperature. pupils use mental recall of addition and subtraction facts to 20 in solving problems involving larger numbers. pupils add and subtract numbers with two digits mentally. pupils add and subtract numbers with three digits using written methods. pupils use mental recall of 2,3,4,5 & 10 multiplication tables and derive division facts pupils solve whole number problems involving x &
3b
3c